Towards Intrinsically Calibrated Uncertainty Quantification in Industrial Data-Driven Models via Diffusion Sampler

Yiran Ma; Jerome Le Ny; Zhichao Chen; Zhihuan Song

cs.LGApr 2, 2026

Towards Intrinsically Calibrated Uncertainty Quantification in Industrial Data-Driven Models via Diffusion Sampler

Authors: Yiran Ma, Jerome Le Ny, Zhichao Chen, Zhihuan Song

Organizations: College of Control Science and Engineering, Zhejiang University, Hangzhou, 310027, China · Electrical Engineering Department, Polytechnique Montreal, and with GERAD, Montreal, QC H3T 1J4, Canada · State Key Lab of General AI, School of Intelligence Science and Technology, Peking University, Beijing 100871, China · Guangdong Provincial Key Laboratory of Petrochemical Equipment Fault Diagnosis, Guangdong University of Petrochemical Technology, Maoming, 525000, China, and also with the College of Control Science and Engineering, Zhejiang University, Hangzhou, 310027, China

Abstract

In modern process industries, data-driven models are important tools for real-time monitoring when key performance indicators are difficult to measure directly. While accurate predictions are essential, reliable uncertainty quantification (UQ) is equally critical for safety, reliability, and decision-making, but remains a major challenge in current data-driven approaches. In this work, we introduce a diffusion-based posterior sampling framework that inherently produces well-calibrated predictive uncertainty via faithful posterior sampling, eliminating the need for post-hoc calibration. In extensive evaluations on synthetic distributions, the Raman-based phenylacetic acid soft sensor benchmark, and a real ammonia synthesis case study, our method achieves practical improvements over existing UQ techniques in both uncertainty calibration and predictive accuracy. These results highlight diffusion samplers as a principled and scalable paradigm for advancing uncertainty-aware modeling in industrial applications.

Figures & tables

Fig. 1 : Overview of the DiffUQ Framework

	$\displaystyle(\theta_{1},c_{1})\leftarrow$	$\displaystyle(\theta_{0},c_{0})+\Sigma_{t=0}^{T}f\!\big(t,[\theta_{t},c_{t}]\big)\,\Delta t$		(11)
		$\displaystyle+\Sigma_{t=0}^{T}g\!\big(t,[\theta_{t},c_{t}]\big)\,\Delta W_{t}$		(11)

Algorithm 1 Training a diffusion sampler

Panel of Figure 3 — Fig. 3 : Comparison on the smiley-face (left) and funnel (right) distributions. (a,e) show mean-field variational contours; other panels show samples from different sampling methods.

Fig. 4 : Summary of the first principle-based mathematical simulator of industrial-scale penicillin simulation [ 39 ]

	Uncertainty Quality			Accuracy
Method	NLL	ECE	MCE	MSE	MAE	$R^{2}$
MAP	-2.6339 ± 0.2131	0.1331 ± 0.0896	0.2267 ± 0.1564	524.7643 ± 86.7733	18.1418 ± 1.4578	0.9020 ± 0.0162
DE	-2.8330 ± 0.0085	0.0654 ± 0.0044	0.1058 ± 0.0068	493.4771 ± 6.1154	17.6894 ± 0.0907	0.9078 ± 0.0011
MC Dropout	-2.3574 ± 0.4713	0.1421 ± 0.1027	0.2537 ± 0.1963	1414.1550 ± 1809.8815	26.3602 ± 16.5030	0.7358 ± 0.3381
MFVI	-2.8593 ± 0.0238	0.0493 ± 0.0071	0.0856 ± 0.0111	486.7107 ± 26.2902	17.5233 ± 0.4535	0.9091 ± 0.0049
SGLD	-2.4741 ± 0.0271	0.2036 ± 0.0080	0.3630 ± 0.0128	541.3067 ± 7.5517	18.5968 ± 0.1380	0.8989 ± 0.0014

TABLE I : Results of Raman-based PAA soft sensor.

Fig. 5 : High-Low Transformer unit from an ammonia synthesis process.

		Uncertainty Quality			Accuracy
Method	$n$	NLL	ECE	MCE	MSE	MAE	$R^{2}$
MAP	1	-0.2729 ± 0.0841	0.0321 ± 0.0269	0.0649 ± 0.0469	0.0330 ± 0.0034	0.1440 ± 0.0079	0.9430 ± 0.0059
DE	4	-0.3363 ± 0.0216	0.0210 ± 0.0104	0.0404 ± 0.0156	0.0298 ± 0.0015	0.1365 ± 0.0040	0.9472 ± 0.0026
	64	-0.3399 ± 0.0204	0.0227 ± 0.0101	0.0450 ± 0.0210	0.0295 ± 0.0015	0.1363 ± 0.0037	0.9489 ± 0.0036
	128	-0.3290 ± 0.0387	0.0229 ± 0.0171	0.0480 ± 0.0303	0.0301 ± 0.0028	0.1376 ± 0.0063	0.9479 ± 0.0039
MC Dropout	4	-0.0038 ± 0.0100	0.0758 ± 0.0017	0.1364 ± 0.0049	0.0602 ± 0.0017	0.1920 ± 0.0021	0.8959 ± 0.0029

TABLE II : Uncertainty qualities and accuracy of the HLT residual

\mathrm{CO}

concentration model based on our method compared with commonly used UQ approaches in industrial models.

Panel of Figure 8 — Fig. 6 : Sensitivity Analysis. The shaded area indicates $\pm 1$ standard deviation over 5 independent runs.

Panel of Figure 9 — Fig. 7 : Training Curves

Panel of Figure 10 — Fig. 8 : Impact of Drift Network Size

Uncertainty Quantification

Uncertainty quantification estimates how reliable predictions and scientific inferences are, and how uncertainty propagates into decisions.

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Diffusion Posterior Sampling with Uncertain Forward Models

Wentao Shangguan, Vivek K Goyal

Department of Electrical and Computer Engineering, Boston University, Boston, MA 02215, USA

Diffusion-Based Inverse Problems Diffusion Posterior Sampling Inverse Problems Bayesian Inverse Problems Uncertainty Propagation

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Diffusion posterior samplers reconstruct signals by combining a pretrained diffusion prior with a measurement likelihood specified by a forward model and a noise model. Calibration and reference data can characterize uncertainty in the forward model's parameters, such as blur kernels or detector responses. Fixing these parameters at estimated values leaves their uncertainty out of the likelihood, and parameter errors are thus absorbed into the reconstruction. We introduce an uncertainty-aware likelihood (UAL), a Gaussian approximation to the marginal likelihood obtained by integrating over the uncertain parameters. For each candidate signal, UAL propagates parameter uncertainty into a structured measurement covariance, reducing the quadratic penalty along residual directions that parameter variation can explain. Existing samplers can use UAL in their data updates while retaining their pretrained diffusion priors and sampling schedules. Under a fixed Gaussian nuisance model, we bound likelihood error from linearization and average divergence between the exact marginal and UAL posteriors. We compare samplers with and without UAL across six inverse problems spanning Gaussian and motion deblurring, CT, MRI, ultrasound, and fluorescence microscopy, and find improvements in reconstruction and ensemble prediction. UAL reduces geometric mean MSE by 9% to 20% across samplers in the main Gaussian-deblurring setting and by 51% to 92% in motion deblurring when non-blind samplers use the population-mean kernel. In motion deblurring, DAPS and PnP-DM with UAL also give 6.2 to 6.4 dB higher ensemble-mean PSNR than published BlindDPS and Fast Diffusion EM. On the ablation set, UAL outperforms trace-matched isotropic inflation for PnP-DM in both deblurring tasks; in motion deblurring, it also exceeds the best-tested fixed scalar data weighting by 3.3 dB for PnP-DM and 4.2 dB for DAPS in average PSNR.

Abstract

Uncertainty Quantification

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